I have a polyline and I want to simplify it with Douglas-Peucker. It's an easy algorithm, but what if there are points with the same distance. What point should I select? Is there a reasonable solution?
I would consider the curvature in that case too. If the curvature is small - then it is a flat region - so you can safely remove it - your 4th point for example. If the curvature is large (your 5th point for example), even if the distance is the same, you should most likely leave it alone.
I would actually recommend a curvature based method coming before Douglas-Peucker.